%!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: contrib2.dvi %%CreationDate: Wed Oct 20 15:21:18 1999 %%Pages: 2 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: D:\EMTEX\BIN\dvips32.exe -pj=tmp.mfj -K contrib2 %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.10.20:1514 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed 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b(of)g(c)n(ars)g(ar)n (e)f(instantane)n(ous)i(ac)n(c)n(or)n(ding)f(to)g(the)g(r)n(outing)g (matrix)f Fq(P)j Fp(=)12 b(\()p Fq(p)1299 542 y Fo(ij)1328 536 y Fp(\))p Fr(.)78 653 y Fn(1.)21 b(In)o(tro)q(duction)-135 739 y Fp(The)14 b(presen)o(t)i(pap)q(er)e(giv)o(es)g(a)f(description)h (of)f(the)i(limit)c(b)q(eha)o(vior)i(of)g(transp)q(ort)i(net)o(w)o (orks)f(with)g(an)f(in\014nite)g(n)o(um)o(b)q(er)g(of)h(no)q(des.)-135 789 y(Pro)q(ofs)j(are)g(omitted)f(due)h(to)g(the)g(lac)o(k)f(of)g (space.)28 b(The)17 b(classi\014cation)f(obtained)h(is)f(not)h (complete)f(and)h(there)h(are)f(still)f(some)-135 838 y(in)o(teresting)f(op)q(en)f(problems)f(to)h(solv)o(e.)-28 900 y(Consider)i(the)h(follo)o(wing)c(queueing)j(system:)22 b(m)o(utually)13 b(indep)q(enden)o(t)k(P)o(oisson)f(arriv)n(al)f(pro)q (cesses)j(with)e(in)o(tensities)g Fq(\025)1930 906 y Fo(i)1944 900 y Fp(,)-124 929 y Fm(1)-130 937 y Fl(P)-135 1005 y Fo(i)p Fk(=1)-74 968 y Fq(\025)-50 974 y Fo(i)-24 968 y Fp(=)11 b Fq(\025)h Fj(\024)g(1)e Fp(en)o(ter)h(a)f(transp)q(ort) h(net)o(w)o(ork)g(with)f(an)g(in\014nite)g(n)o(um)o(b)q(er)f(of)h (stations.)17 b(There)11 b(are)g Fq(x)1467 974 y Fo(i)1491 968 y Fp(cars)g(in)f(an)g(arbitrary)g(no)q(de)h Fq(i)-135 1036 y Fl(\000)-105 1030 y Fm(1)-111 1038 y Fl(P)-116 1106 y Fo(i)p Fk(=1)-55 1069 y Fq(x)-31 1075 y Fo(i)-6 1069 y Fp(=)h Fj(1)80 1036 y Fl(\001)109 1069 y Fp(at)f(the)g(initial)e (momen)o(t)f Fq(t)k Fp(=)g(0)e(whic)o(h)h(instan)o(tly)f(transp)q(ort)i (clien)o(ts)f(to)g(di\013eren)o(t)h(no)q(des)f(according)g(to)g(the)g (in\014nite-)-135 1145 y(dimensional)k(routing)h(matrix)f Fi(P)23 b Fp(=)17 b(\()p Fq(p)522 1151 y Fo(ij)552 1145 y Fp(\))568 1130 y Fm(1)568 1156 y Fo(i;j)r Fk(=1)665 1145 y Fp(:)g(if)f(a)g(clien)o(t)h(en)o(ters)i(a)d(no)q(de)h Fq(i)h Fp(with)e(an)h(a)o(w)o(aiting)e(v)o(ehicle)i(the)g(car)h(instan) o(tly)-135 1195 y(carries)13 b(the)g(customer)f(to)f(a)h(station)g Fq(j)i Fp(with)d(probabilit)o(y)g Fq(p)814 1201 y Fo(ij)843 1195 y Fq(:)g Fp(If)h(there)h(are)f(no)g(cars)g(presen)o(t)i(in)d(the)i (no)q(de)f Fq(i)g Fp(\(w)o(e)g(shall)f(call)h(suc)o(h)g(a)-135 1245 y(no)q(de)g(the)f Fr(empty)h Fp(one\))g(the)f(customer)g(lea)o(v)o (es)g(the)g(system)g(\(there)h(are)g(no)e(accum)o(ulators)g(for)h (customers\);)h(accum)o(ulators)e(for)g(cars)-135 1316 y(are)k(of)f(in\014nite)g(capacit)o(y)m(.)k(Assume)c(that)h Fj(8)p Fq(i)e Fj(2)f Fp(I)-7 b(N)700 1276 y Fm(1)694 1284 y Fl(P)687 1352 y Fo(j)r Fk(=1)752 1316 y Fq(p)773 1322 y Fo(ij)813 1316 y Fp(=)12 b(1)p Fq(;)20 b(p)931 1322 y Fo(ii)968 1316 y Fp(=)11 b(0.)18 b(If)13 b(the)h(con)o(trary)g (is)f(not)g(stated)h(supp)q(ose)h(that)e Fq(\025)f(<)g Fj(1)p Fp(.)-135 1403 y(Consider)17 b(the)g(space)g Fh(Z)-18 b(Z)260 1387 y Fk(+)285 1375 y Fg(1)335 1403 y Fp(where)17 b Fh(Z)-18 b(Z)489 1384 y Fk(+)532 1403 y Fp(=)16 b(I)-7 b(N)10 b Fj([)f(f)p Fp(0)p Fj(g)o Fp(.)25 b(In)o(tro)q(duce)17 b(the)g(pro)q(duct)g Fq(\033)q Fj(\000)p Fp(algebra)f Fj(B)i Fp(on)e Fh(Z)-18 b(Z)1529 1387 y Fk(+)1554 1375 y Fg(1)1587 1403 y Fp(.)25 b(Let)17 b Fj(8)p Fq(i)g(X)1789 1409 y Fo(i)1803 1403 y Fp(\()p Fq(n)p Fp(\))f(b)q(e)h(a)-135 1457 y(n)o(um)o(b)q(er)d(of)g(cars)h(in)e(the)i(no)q(de)g Fq(i)f Fp(at)g(the)h(momen)o(t)d(of)i Fq(n)p Fp(-th)g(arriv)n(al)f(to)h (the)h(system.)j(Denote)1384 1447 y Fq(~)1375 1457 y(X)t Fp(\()p Fq(n)p Fp(\))12 b(=)1527 1424 y Fl(\000)1546 1457 y Fq(X)1580 1463 y Fk(1)1599 1457 y Fp(\()p Fq(n)p Fp(\))p Fq(;)h(:)7 b(:)g(:)12 b(X)1777 1463 y Fo(i)1792 1457 y Fp(\()p Fq(n)p Fp(\))p Fq(;)h(:)7 b(:)g(:)1937 1424 y Fl(\001)-135 1516 y Fj(8)p Fq(n)13 b Fj(2)f Fp(I)-7 b(N)p Fq(:)14 b Fp(Let)106 1482 y Fl(\000)125 1516 y Fp(\012)p Fq(;)7 b Fj(L)p Fq(;)g(P)255 1482 y Fl(\001)287 1516 y Fp(b)q(e)15 b(a)f(giv)o(en)g(probabilit)o(y)f(space.)20 b(Then)947 1506 y Fq(~)938 1516 y(X)t Fp(\()p Fq(n)p Fp(\))13 b(:)1077 1482 y Fl(\000)1096 1516 y Fp(\012)p Fq(;)7 b Fj(L)p Fq(;)g(P)1226 1482 y Fl(\001)1255 1516 y Fj(!)1309 1482 y Fl(\000)1328 1516 y Fn(Z)1357 1501 y Fk(+)1382 1488 y Fg(1)1415 1516 y Fq(;)g Fj(B)1462 1482 y Fl(\001)1496 1516 y Fp(is)14 b(a)g(sto)q(c)o(hastic)i(pro)q (cess.)21 b(If)-135 1575 y(w)o(e)14 b(denote)h Fq(P)87 1582 y Fo(~)-17 b(x)108 1575 y Fp(\()p Fj(\001)p Fp(\))11 b(=)h Fq(P)6 b Fj(f\001)k(j)315 1564 y Fq(~)306 1575 y(X)t Fp(\(0\))h(=)h Fq(~)-21 b(x)p Fj(g)14 b(8)o Fq(~)-20 b(x)11 b Fj(2)g Fh(Z)-18 b(Z)640 1560 y Fk(+)665 1547 y Fg(1)712 1575 y Fp(then)15 b Fi(X)g Fp(=)905 1541 y Fl(\000)932 1564 y Fq(~)924 1575 y(X)s Fp(\()p Fq(n)p Fp(\))p Fq(;)f(P)1071 1582 y Fo(~)-17 b(x)1092 1541 y Fl(\001)1124 1575 y Fp(app)q(ears)15 b(to)f(b)q(e)g(a)g(non-coun)o (table)g(Mark)o(o)o(v)f(c)o(hain.)78 1697 y Fn(2.)21 b(De\014nition)o(s)13 b(and)i(results)-28 1783 y Fp(D)6 b(e)g(f)g(i)g(n)g(i)g(t)g(i)g(o)g(n)28 b(2.1.)22 b Fr(The)17 b(network)f(\(the)g(pr)n(o)n(c)n(ess)g Fi(X)t Fr(\))g(is)g(c)n(al)r(le) n(d)h Fp(ergo)q(dic)f Fr(if)g(the)h(se)n(quenc)n(e)g(of)f(me)n(asur)n (es)h Fq(P)1729 1768 y Fo(n)1723 1795 y(~)-17 b(x)1767 1783 y Fr(which)16 b(c)n(or-)-135 1840 y(r)n(esp)n(ond)h(to)g(the)g (distribution)f(of)g Fi(X)21 b Fr(on)c(every)g(step)f Fq(n)h Fr(c)n(onver)n(ges:)23 b Fq(P)1016 1825 y Fo(n)1010 1852 y(~)-17 b(x)1038 1840 y Fp(\(\000\))15 b(=)g Fq(P)1185 1847 y Fo(~)-17 b(x)1205 1840 y Fj(f)1235 1829 y Fq(~)1226 1840 y(X)t Fp(\()p Fq(n)p Fp(\))15 b Fj(2)f Fp(\000)p Fj(g)h(\000)-7 b(!)15 b Fq(\026)7 b Fp(\(\000\))22 b Fr(as)15 b Fq(n)f Fj(!)h(1)42 b(8)o Fq(~)-20 b(x)14 b Fj(2)-135 1912 y Fh(Z)-18 b(Z)-103 1897 y Fk(+)-78 1885 y Fg(1)-21 1912 y Fp(:)32 1873 y Fm(1)26 1881 y Fl(P)21 1949 y Fo(i)p Fk(=1)82 1912 y Fq(x)106 1918 y Fo(i)143 1912 y Fp(=)24 b Fj(1)p Fq(;)d Fj(8)7 b Fp(\000)23 b Fj(2)g(B)j Fp(:)21 b(\000)g Fr(is)g(a)h(set)f(of)h Fq(\026)f Fr({)h(c)n(ontinuity,)h(wher)n(e)e Fq(\026)h Fr(is)f(a)h(pr)n(ob)n (ability)e(me)n(asur)n(e)i(on)1768 1879 y Fl(\000)1787 1912 y Fh(Z)-18 b(Z)1819 1897 y Fk(+)1844 1885 y Fg(1)1877 1912 y Fq(;)7 b Fj(B)1924 1879 y Fl(\001)1943 1912 y Fr(,)-135 2017 y Fq(\026)-103 1959 y Fl(\022)-73 2017 y Fq(~)-20 b(x)11 b Fp(:)3 1978 y Fm(1)-2 1986 y Fl(P)-7 2054 y Fo(i)p Fk(=1)54 2017 y Fq(x)78 2023 y Fo(i)103 2017 y Fp(=)h Fj(1)189 1959 y Fl(\023)230 2017 y Fp(=)g(1)p Fq(:)-135 2155 y Fp(Denote)g Fq(\015)26 2161 y Fo(i)52 2155 y Fp(=)g Fq(\025)120 2161 y Fo(i)134 2155 y Fq(=\025)22 b Fj(8)p Fq(i)12 b Fj(2)f Fp(I)-7 b(N)p Fq(;)368 2115 y Fm(1)362 2124 y Fl(P)358 2191 y Fo(i)p Fk(=1)418 2155 y Fq(\015)439 2161 y Fo(i)465 2155 y Fp(=)12 b(1)p Fq(:)e Fp(Supp)q(ose)i(the)g(Mark)o(o)o(v)e(c)o(hain)h(with)g(the)g(state)h (space)h(I)-7 b(N)11 b(and)g(the)g(transition)g(matrix)e Fi(P)-135 2256 y Fp(to)14 b(b)q(e)h(homogeneous,)d(ap)q(erio)q(dic,)i (and)g(irreducible.)19 b(In)o(tro)q(duce)c Fq(G)p Fp(\()p Fq(i;)7 b(j)r Fp(\))12 b(=)1109 2216 y Fm(1)1103 2225 y Fl(P)1094 2292 y Fo(n)p Fk(=0)1164 2256 y Fq(p)1185 2234 y Fk(\()p Fo(n)p Fk(\))1185 2267 y Fo(ij)1261 2256 y Fj(8)p Fq(i;)7 b(j)14 b Fj(2)d Fp(I)-7 b(N)14 b(where)i Fq(p)1582 2234 y Fk(\()p Fo(n)p Fk(\))1582 2267 y Fo(ij)1644 2256 y Fp(is)e(the)g(probabilit)o(y)-135 2329 y(to)g(reac)o(h)h(the)f (no)q(de)h Fq(i)f Fp(from)e(the)i(no)q(de)h Fq(j)h Fp(in)d Fq(n)h Fp(steps.)-28 2415 y(D)6 b(e)g(f)g(i)g(n)g(i)g(t)g(i)g(o)g(n)27 b(2.2.)21 b Fr(Matrix)16 b Fi(P)k Fp(=)14 b(\()p Fq(p)593 2421 y Fo(ij)622 2415 y Fp(\))638 2421 y Fo(i;j)r Fm(2)p Fk(I)-6 b(N)747 2415 y Fr(is)16 b(c)n(al)r(le)n(d)f(p)n(ositive)h(r)n (e)n(curr)n(ent)f(or)h(er)n(go)n(dic)g(\(r)n(esp.)22 b(nul)r(l)16 b(-)f(r)n(e)n(curr)n(ent,)g(tr)n(an-)-135 2465 y(sient\))g(if)f(a)h(c)n(ountable)h(Markov)f(chain)h(with)e(tr)n (ansition)g(matrix)g Fi(P)22 b Fr(is)14 b(p)n(ositive)h(r)n(e)n(curr)n (ent)f(\(r)n(esp.)k(nul)r(l)d(-)g(r)n(e)n(curr)n(ent,)e(tr)n (ansient\).)-135 2551 y Fp(In)f(the)h(case)f(of)f(p)q(ositiv)o(e)h (recurrence)j(there)e(exists)f(suc)o(h)h(a)f(probabilit)o(y)e (distribution)h Fq(\031)i Fp(=)1321 2518 y Fl(\000)1340 2551 y Fq(\031)1364 2557 y Fo(i)1378 2518 y Fl(\001)1397 2568 y Fo(i)p Fm(2)p Fk(I)-6 b(N)1476 2551 y Fp(\(later)12 b(on)f(called)h Fr(stationary)p Fp(\))-135 2630 y(that)i Fj(8)p Fq(i)e Fj(2)f Fp(I)-7 b(N)28 b Fq(\031)134 2636 y Fo(i)159 2630 y Fp(=)215 2591 y Fm(1)210 2599 y Fl(P)203 2667 y Fo(j)r Fk(=1)267 2630 y Fq(\031)291 2636 y Fo(j)309 2630 y Fq(p)330 2636 y Fo(j)r(i)373 2630 y Fp(\(brie\015y)m(,)12 b Fq(~)-20 b(\031)13 b Fp(=)d Fq(~)-20 b(\031)r Fi(P)20 b Fp(\).)-28 2748 y(D)6 b(e)g(f)g(i)g(n)g(i)g(t)g(i)g(o)g(n)33 b(2.3.)c Fr(Me)n(asur)n(e)18 b Fq(\026)p Fp(\()p Fj(\001)p Fp(\))h Fr(on)g Fp(I)-7 b(N)19 b Fr(\(p)n(ossibly)f Fq(\033)q Fr(-\014nite\))h(is)g(c)n(al)r(le)n(d)f(\(strictly\))f(exc)n(essive)i (for)f(tr)n(ansition)g(matrix)-135 2819 y Fi(P)h Fp(=)11 b(\()p Fq(p)-6 2825 y Fo(ij)24 2819 y Fp(\))40 2825 y Fo(i;j)r Fm(2)p Fk(I)-6 b(N)147 2819 y Fr(i\013)30 b Fq(\026)11 b Fj(\025)h Fq(\026)p Fi(P)18 b Fp(:)g Fj(8)p Fq(i)d(\026)475 2825 y Fo(i)500 2819 y Fj(\025)556 2780 y Fm(1)551 2788 y Fl(P)544 2856 y Fo(j)r Fk(=1)608 2819 y Fq(\026)633 2825 y Fo(j)651 2819 y Fq(p)672 2825 y Fo(j)r(i)716 2819 y Fr(\()p Fq(\026)c(>)h(\026)p Fi(P)21 b Fr(,)15 b(r)n(esp)n(e)n(ctively\).)p eop %%Page: 2 2 2 1 bop 31 -14 a Fp(P)6 b(r)g(o)g(p)g(o)g(s)g(i)g(t)g(i)g(o)g(n)34 b(2.1)14 b([Necessary)j(conditions)e(of)f(ergo)q(dicit)o(y].)22 b Fr(If)16 b(the)g(network)g(is)g(er)n(go)n(dic,)g(then)g(matrix)g Fi(P)22 b Fr(is)16 b(p)n(ositive)-76 36 y(r)n(e)n(curr)n(ent.)31 131 y Fp(P)6 b(r)g(o)g(p)g(o)g(s)g(i)g(t)g(i)g(o)g(n)24 b(2.2)10 b([D.)f(Khmelev,)h(V.)g(Oseledets)j(\(1998\)].)k Fr(If)11 b Fi(P)19 b Fr(is)11 b(p)n(ositive)h(r)n(e)n(curr)n(ent,)1503 92 y Fm(1)1498 100 y Fl(P)1493 168 y Fo(i)p Fk(=1)1558 114 y Fo(\031)1577 118 y Ff(i)p 1558 121 33 2 v 1558 146 a Fo(\025)1578 150 y Ff(i)1608 131 y Fq(<)f Fj(1)p Fr(,)h(then)g(the)g(network)-76 205 y(is)j(not)g(er)n(go)n(dic.)31 290 y Fp(L)10 b(e)g(m)f(m)g(a)27 b(2.1.)18 b Fr(If)c Fi(P)22 b Fr(is)14 b(er)n(go)n(dic)h(and)h Fj(f)p Fq(\015)720 296 y Fo(i)734 290 y Fj(g)755 275 y Fm(1)755 301 y Fo(i)p Fk(=1)822 290 y Fj(6)p Fp(=)c Fj(f)p Fq(\031)911 296 y Fo(i)924 290 y Fj(g)945 275 y Fm(1)945 301 y Fo(i)p Fk(=1)1015 290 y Fr(or)j(if)f Fi(P)22 b Fr(is)14 b(not)h(er)n(go)n(dic) g(then)548 385 y Fj(9)7 b Fq(i)592 391 y Fk(0)611 385 y Fq(;)13 b(j)653 391 y Fk(0)683 385 y Fp(:)54 b Fq(\015)770 391 y Fo(i)782 395 y Fe(0)812 385 y Fq(>)860 345 y Fl(X)856 435 y Fo(j)r Fm(6)p Fk(=)p Fo(i)909 439 y Fe(0)931 385 y Fq(\015)952 391 y Fo(j)970 385 y Fq(p)991 391 y Fo(j)r(i)1019 395 y Fe(0)1037 385 y Fq(;)49 b(\015)1119 391 y Fo(j)1133 395 y Fe(0)1163 385 y Fq(<)1212 345 y Fl(X)1206 435 y Fo(j)r Fm(6)p Fk(=)p Fo(j)1261 439 y Fe(0)1284 385 y Fq(\015)1305 391 y Fo(j)1323 385 y Fq(p)1344 391 y Fo(j)r(j)1374 395 y Fe(0)31 576 y Fp(T)8 b(h)g(e)g(o)g(r)h(e)f(m)36 b(2.1)17 b([Sto)q(c)o(hastical)h(b)q(oundedness].)33 b Fr(If)19 b Fq(\015)950 582 y Fo(i)962 586 y Fe(0)1000 576 y Fq(>)1063 536 y Fm(1)1058 545 y Fl(P)1051 612 y Fo(j)r Fk(=1)1115 576 y Fq(\015)1136 582 y Fo(j)1154 576 y Fq(p)1175 582 y Fo(j)r(i)1203 586 y Fe(0)1259 576 y Fr(for)g(some)g Fq(i)1454 582 y Fk(0)1473 576 y Fr(,)h(then)f Fq(X)1636 582 y Fo(i)1648 586 y Fe(0)1667 576 y Fp(\()p Fj(\001)p Fp(\))g Fr(is)f(sto)n(chastic)n(al)r(ly)-76 655 y(b)n(ounde)n(d.)-76 750 y Fp(F)m(or)e(some)g Fj(f)p Fq(a)151 756 y Fo(i)164 750 y Fj(g)185 735 y Fm(1)185 761 y Fo(i)p Fk(=1)241 750 y Fq(;)48 b(a)323 756 y Fo(i)352 750 y Fj(\025)16 b Fp(0)p Fq(;)491 711 y Fm(1)485 719 y Fl(P)480 787 y Fo(i)p Fk(=1)541 750 y Fq(a)563 756 y Fo(i)592 750 y Fq(<)g Fj(1)g Fp(and)g(for)g(all)e Fq(~)-20 b(x)15 b Fj(2)g Fh(Z)-18 b(Z)1021 735 y Fk(+)1046 723 y Fg(1)1096 750 y Fp(de\014ne)17 b Fq(f)t Fp(\()o Fq(~)-20 b(x)q Fp(\))15 b(=)1372 711 y Fm(1)1367 719 y Fl(P)1362 787 y Fo(i)p Fk(=1)1423 750 y Fq(a)1445 756 y Fo(i)1459 750 y Fq(x)1483 756 y Fo(i)1512 750 y Fj(\024)g(1)p Fp(.)25 b(Let)17 b(us)f(in)o(tro)q(duce)h(the)-76 843 y(family)11 b(of)j Fq(\033)h Fp({)e(algebras)h Fj(F)364 849 y Fo(n)398 843 y Fp(=)e Fq(\033)474 797 y Fl(\020)507 832 y Fq(~)499 843 y(X)s Fp(\()p Fq(m)p Fp(\))p Fq(;)21 b(m)12 b Fj(\024)g Fq(n)754 797 y Fl(\021)813 843 y Fj(8)p Fq(n)7 b(:)31 940 y Fp(T)h(h)g(e)g(o)g(r)h(e)f(m)35 b(2.2)17 b([Non-ergo)q(dic)g (case:)28 b(existence)19 b(of)f(sup)q(ermartingales].)29 b Fr(If)18 b(the)h(matrix)f Fi(P)25 b Fr(is)19 b(tr)n(ansient)f(and)h (ther)n(e)-76 1011 y(exist)g(numb)n(ers)g Fj(f)p Fq(')242 1017 y Fo(i)255 1011 y Fj(g)276 996 y Fm(1)276 1022 y Fo(i)p Fk(=1)350 1011 y Fr(such)h(that)e Fq(')564 1017 y Fo(i)596 1011 y Fj(\025)h Fp(0)p Fq(;)708 972 y Fm(1)703 980 y Fl(P)698 1047 y Fo(i)p Fk(=1)771 972 y Fm(1)765 980 y Fl(P)759 1047 y Fo(j)r Fk(=1)823 1011 y Fq(G)p Fp(\()p Fq(i;)7 b(j)r Fp(\))p Fq(')967 1017 y Fo(j)1003 1011 y Fq(<)19 b Fj(1)p Fr(,)f(then)h(for)g Fq(a)1316 1017 y Fo(i)1348 1011 y Fp(=)1398 977 y Fl(\000)1417 1011 y Fq(G')1477 977 y Fl(\001)1496 1028 y Fo(i)1528 1011 y Fp(=)1591 972 y Fm(1)1586 980 y Fl(P)1579 1047 y Fo(j)r Fk(=1)1643 1011 y Fq(G)p Fp(\()p Fq(i;)7 b(j)r Fp(\))p Fq(')1787 1017 y Fo(j)1823 1011 y Fr(and)20 b(initial)-76 1118 y(distributions)10 b(of)h(Markov)g(chain)g Fi(X)j Fr(such)d(that)692 1078 y Fm(1)687 1086 y Fl(P)682 1154 y Fo(i)p Fk(=1)743 1118 y Fq(a)765 1124 y Fo(i)779 1118 y Fq(X)813 1124 y Fo(i)827 1118 y Fp(\(0\))g Fq(<)h Fj(1)e Fr(a.)18 b(s.)f(and)12 b Fq(E)1213 1078 y Fm(1)1208 1086 y Fl(P)1203 1154 y Fo(i)p Fk(=1)1263 1118 y Fq(a)1285 1124 y Fo(i)1299 1118 y Fq(X)1333 1124 y Fo(i)1347 1118 y Fp(\(0\))g Fq(<)g Fj(1)e Fr(se)n(quenc)n(e)1672 1071 y Fl(\020)1697 1118 y Fq(f)1721 1084 y Fl(\000)1749 1107 y Fq(~)1741 1118 y(X)s Fp(\()p Fq(n)p Fp(\))1835 1084 y Fl(\001)1855 1118 y Fq(;)d Fj(F)1904 1124 y Fo(n)1926 1071 y Fl(\021)1950 1080 y Fm(1)1950 1147 y Fo(n)p Fk(=1)-76 1191 y Fr(wil)r(l)14 b(b)n(e)h(a)g(sup)n(ermartingale,)f(and)i(henc)n (e)f(ther)n(e)g(exists)f(some)h(r)n(andom)h(variable)e Fq(X)1264 1197 y Fm(1)1311 1191 y Fp(:)k Fq(E)r(X)1408 1197 y Fm(1)1455 1191 y Fq(<)12 b Fj(1)i Fr(such)i(that)683 1256 y Fm(1)670 1269 y Fl(X)673 1357 y Fo(i)p Fk(=1)737 1308 y Fq(a)759 1314 y Fo(i)772 1308 y Fq(X)806 1314 y Fo(i)821 1308 y Fp(\()p Fq(n)p Fp(\))889 1281 y Fd(w.p.1)901 1308 y Fj(\000)-7 b(!)23 b Fq(X)1025 1314 y Fm(1)1074 1308 y Fr(as)15 b Fq(n)c Fj(!)g(1)p Fq(:)31 1468 y Fp(C)6 b(o)g(r)g(o)g(l)g(l)g(a)g(r)g(y)35 b(2.1)16 b([Sto)q(c)o(hastical)h(b)q (oundedness].)29 b Fr(Under)17 b(c)n(onditions)i(of)e(the)n(or)n(em)h (2.2)g(al)r(l)f(no)n(des)h(of)g(the)g(network)f(ar)n(e)-76 1517 y(sto)n(chastic)n(al)r(ly)e(b)n(ounde)n(d.)31 1621 y Fp(P)6 b(r)g(o)g(p)g(o)g(s)g(i)g(t)g(i)g(o)g(n)40 b(2.3.)30 b Fr(If)19 b Fj(9)7 b Fq(j)531 1627 y Fk(0)569 1621 y Fp(:)617 1581 y Fm(1)612 1590 y Fl(P)607 1657 y Fo(i)p Fk(=1)667 1621 y Fq(G)p Fp(\()p Fq(i;)g(j)766 1627 y Fk(0)785 1621 y Fp(\))19 b Fq(<)g Fj(1)p Fr(,)h(then)f(ther)n(e)g (exist)g(such)g(numb)n(ers)g Fj(f)p Fq(')1566 1627 y Fo(i)1580 1621 y Fj(g)1601 1606 y Fm(1)1601 1632 y Fo(i)p Fk(=1)1676 1621 y Fp(:)32 b Fq(')1747 1627 y Fo(i)1781 1621 y Fj(\025)19 b Fp(0)g Fr(that)g(the)-76 1694 y(c)n(onditions)d(of) f(the)n(or)n(em)f(2.2)h(hold.)31 1779 y Fp(R)8 b(e)h(m)e(a)h(r)h(k)29 b(2.1.)21 b Fr(L)n(emma)16 b(2.1)f(states)h(that)g(strictly)e(exc)n (essive)i(pr)n(ob)n(ability)f(me)n(asur)n(es)g Fq(\015)k Fr(do)d(not)g(exist)f(for)h Fi(P)22 b Fr(if)15 b Fq(\025)e(<)h Fj(1)p Fr(.)-76 1829 y(But,)h(if)g Fq(\025)c Fp(=)h Fj(1)j Fr(and)g Fi(P)22 b Fr(is)14 b(tr)n(ansient,)g(then)i(ther)n(e)e(exist)h (in\014nitely)g(many)g Fq(\033)q Fj(\000)p Fr(\014nite)h(strictly)d (exc)n(essive)j(me)n(asur)n(es)e Fq(\026)e Fp(:)-65 1858 y Fm(1)-71 1866 y Fl(P)-76 1934 y Fo(i)p Fk(=1)-15 1898 y Fq(\026)10 1904 y Fo(i)39 1898 y Fp(=)k Fj(1)p Fq(;)49 b(\026)15 b(>)h(\026)p Fn(P)p Fq(:)g Fr(Henc)n(e)i(if)e(we)h(de\014ne)h (our)f(pr)n(o)n(c)n(ess)g Fj(f)p Fq(X)996 1904 y Fo(i)1010 1898 y Fp(\()p Fq(t)p Fp(\))f(:)22 b Fq(i)16 b Fj(2)f Fp(I)-7 b(N)p Fq(;)49 b(t)16 b Fj(\025)f Fp(0)p Fj(g)i Fr(pr)n(op)n(erly)f(in)h(c)n(ase)h Fq(\025)e Fp(=)f Fj(1)i Fr(then)h(for)-76 1973 y Fj(f)p Fq(\025)-31 1979 y Fo(i)-17 1973 y Fj(g)4 1958 y Fm(1)4 1984 y Fo(i)p Fk(=1)71 1973 y Fp(:)g Fq(\025)125 1979 y Fo(i)151 1973 y Fq(>)199 1942 y Fl(P)195 2011 y Fo(j)r Fm(6)p Fk(=)p Fo(i)254 1973 y Fq(\025)278 1979 y Fo(j)296 1973 y Fq(p)317 1979 y Fo(j)r(i)389 1973 y Fj(8)p Fq(i)11 b Fj(2)h Fp(I)-7 b(N)15 b Fr(al)r(l)f(c)n(o)n(or)n(dinates)h Fq(X)840 1979 y Fo(i)854 1973 y Fp(\()p Fq(t)p Fp(\))g Fr(wil)r(l)f(b)n(e)h(sto) n(chastic)n(al)r(ly)f(b)n(ounde)n(d.)137 2124 y Fn(Ac)o(kno)o (wledgemen)o(ts)-76 2210 y Fc(The)e(author)d(would)i(like)f(to)h(thank) e(Pr)n(of.)17 b(Afanas'ev)o(a)9 b(\(Mosc)n(ow)h(State)g(University\))e (and)i(Pr)n(of.)17 b(Engelb)n(ert)9 b(\(F)m(rie)n(drich)g(Schil)r(ler)h (University)-76 2255 y(of)j(Jena\))g(for)g(helpful)f(c)n(omments)g(and) g(discussions.)137 2368 y Fn(3.)21 b(References)-69 2463 y Fd(1)13 b Fb(F)l(a)n(yolle,)j(G.,)i(Lasgouttes,)f(G.)g(-)h(M.)p Fd(:)k(Asymptotics)c(and)e(scalings)j(for)c(large)i(pro)q(duct-form)g (net)o(w)o(orks)g(via)g(the)f(Cen)o(tral)h(Limit)-37 2509 y(Theorem;)c(\(1996\),)g(Preprin)o(t.)-69 2555 y(2)g Fb(W)l(alrand,)i(J.)p Fd(:)h(An)d(in)o(tro)q(duction)j(to)d(queueing)i (net)o(w)o(orks;)e(Pren)o(tice)h(Hall,)g(\(1988\).)-69 2600 y(3)f Fb(W)l(a)n(ymire,)g(E.)p Fd(:)j(Zero{range)e(in)o(teraction) g(at)e(Bose)h(|)g(Eistein)h(sp)q(eeds)f(under)g(a)f(p)q(ositiv)o(e)j (recurren)o(t)d(single)j(particle)f(la)o(w;)e(Ann.)17 b(Prob.,)-37 2646 y Fa(8)p Fd(,)12 b(N3,)h(\(1980\),)g(441)g({)g(450.) -69 2692 y(4)g Fb(Liggett,)h(T.)g(M.)p Fd(:)j(In)o(teracting)d (particle)h(systems;)e(Springer,)h(1985.)-76 2791 y Fr(A)n(ddr)n(ess:) 19 b Ft(Eugene)d(Spod)o(arev)i Fp(,)13 b(F)m(riedric)o(h)h(Sc)o(hiller) g(Univ)o(ersit)o(y)m(,)e(Institute)j(of)e(Sto)q(c)o(hastics,)i(Ernst)g (Abb)q(e)g(Platz)f(1-4,)e(Jena)96 2841 y(07743,)g(German)o(y)m(,)f (E-mail:)16 b(SEU@minet.uni-jena.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF